Permutation Puzzles: A Mathematical Perspective

MATH 301: Mathematical Journeys I

Instructor: Jamie MulhollandTA: Jas Dahan

Course Webpage: https://www.sfu.ca/~jtmulhol/permutationpuzzles

Note: page redirects to http://www.sfu.ca/~jtmulhol/math302/

Grading Scheme: Important Dates:

Assignments 30%Midterm 1 25%Midterm 2 25%Final Poster 20%

Midterm 1 Wed. Oct. 9 Midterm 2 Fri. Nov. 15 Final Poster Dec. 10 or 11

You can check your grades on Canvas.

Textbook:Available for download from course webpage.

Why Study Puzzles?• Puzzles are the common persons experience with abstraction, and math is all about abstraction• Provides a wonderful hands-on approach to group theory (an important branch of mathematics)• They help to illustrate and reinforce the fundamental technique of problem solving:

if you don’t know how to solve a problem, find a simpler one you do know how to solve.• Puzzles can be FUN

Why Study Group Theory?

• Conservation laws of physics are related to the symmetry of physical laws under various transformations.

• Modern particle physics would not exist without group theory; in fact, group theory predicted the existence of many elementary particles before they were found experimentally.

Broadly speaking, group theory is the study of symmetry.We apply the label symmetry to anything which stays invariant under some transformation.

• The structure and behaviour of molecules and crystals depends on their different symmetries.

Chemistry

Physics

Why Study Group Theory?

• Group theory is very closely linked to symmetry in geometry.• Classical problems in algebra have been resolved with

group theory.(no general solution to a degree 5 polynomial equation, unlike the degree 2 case where we have the quadratic formula)

Broadly speaking, group theory is the study of symmetry.We apply the label symmetry to anything which stays invariant under some transformation.

Mathematics

• The mathematics of public-key cryptography uses group theory.• this is what is used when doing online purchases

• Identification numbers are all around us:• ISBN number for a book• VIN (Vehicle Identification Number) for your car• Bar code on a UPS package• Credit card numbers

Day to day life

Why Study Group Theory?

• With group theory we can completely understand Rubiks’ Cube, and other puzzles.• The 15-puzzle drove the world crazy in the 1880’s and group theory provided the cure.• Science fiction shows (Futurama, Stargate SG1) feature problems requiring group theory.

Broadly speaking, group theory is the study of symmetry.We apply the label symmetry to anything which stays invariant under some transformation.

For Fun

The 15-Puzzle Oval Track (TopSpin) Hungarian Rings

Rubik’s CubeLights Out

Jaap’s Puzzle Page: https://www.jaapsch.net/puzzles/

Rubik’s Cube

Ideal Toy Company stated on the package of the original Rubik cube that there were more than three billion possible states the cube could attain. It's analogous to McDonald's proudly announcing that they've sold more than 120 hamburgers.(J. A. Paulos, Innumeracy)

There are actually

8!·12!·37·210 = 43,252,003,274,489,856,000 configurations of the cube.

Some questions we will answer in this course:

Are all configurations of the puzzle pieces solvable?

If not, can we determine… … a criteria for which puzzles are solvable? … the number of solvable configurations? … the probability that a random configuration of the pieces is solvable?

Can we find a strategy for solving the puzzle?

Why do we know there exist configurations of Rubik’s Cube that no cube has ever been twisted into?

Why is it impossible to flip only one edge sub-cube of Rubik’s Cube?

What is the maximum number of moves required to solve Rubik’s Cube?

How can we use computational tools to help us understand and solve this puzzles?

We will not be interested in “speedsolving” Rubik’s cube. Our attention is on understanding the cube. Any solution strategies we use for solving the cube we will discover. We won’t be focussing on any of the algorithms which can be found on the web.

Note:

SageMath: a free open-source mathematics software package

We will use SageMath in two ways:

(1) through a SageMath server: https://sage.syzygy.ca (requires a GitHub account - which is free)

(2) SageMath cell server (for quick calculations):

Appendix A in the course text has all the information needed to get up and running with SageMath.

Lights OutSample game play:

http://www.highcaffeinecontent.com/lightsoff/

http://www.sfu.ca/~jtmulhol/math302/puzzles-lo.html

Homework: Play Lights Out